Self-organized Critical Model Research Articles

1/fα noise in the Robin Hood model

Abstract We consider the Robin Hood dynamics, a one-dimensional extremal self-organized critical model that describes the low-temperature creep phenomenon. One of the key quantities is the time evolution of the state variable (force noise). To understand the temporal correlations, we compute the power spectra of the local force fluctuations and apply finite-size scaling to get scaling functions and critical exponents. We find a signature of the 1 / f α noise for the local force with a nontrivial value of the spectral exponent 0 < α < 2 . We also examine temporal fluctuations in the position of the extremal site and a local activity signal. We present results for different local interaction rules of the model.

Explanation of flicker noise with the Bak-Tang-Wiesenfeld model of self-organized criticality.

With the original Bak-Tang-Wisenefeld (BTW) sandpile we uncover the 1/φ noise in the mechanism maintaining self-organized criticality (SOC)-the question raised together with the concept of SOC. The BTW sandpile and the phenomenon of SOC in general are built on the slow time scale at which the system is loaded and the fast time scale at which the stress is transported outward from overloaded locations. Exploring the dynamics of stress in the slow time in the BTW sandpile, we posit that it follows cycles of gradual stress accumulation that end up with an abrupt stress release and the drop of the system to subcritical state. As the system size grows, the intracycle dynamics exhibits the 1/φ-like spectrum that extends boundlessly and corresponds to the stress release within the critical state.

Q factor: A measure of competition between the topper and the average in percolation and in self-organized criticality.

We define the Q factor in the percolation problem as the quotient of the size of the largest cluster and the average size of all clusters. As the occupation probability p is increased, the Q factor for the system size L grows systematically to its maximum value Q_{max}(L) at a specific value p_{max}(L) and then gradually decays. Our numerical study of site percolation problems on the square, triangular, and simple cubic lattices exhibits that the asymptotic values of p_{max}, though close, are distinct from the corresponding percolation thresholds of these lattices. We also show, using scaling analysis, that at p_{max} the value of Q_{max}(L) diverges as L^{d} (d denoting the dimension of the lattice) as the system size approaches its asymptotic limit. We further extend this idea to nonequilibrium systems such as the sandpile model of self-organized criticality. Here the Q(ρ,L) factor is the quotient of the size of the largest avalanche and the cumulative average of the sizes of all the avalanches, with ρ the drop density of the driving mechanism. This study was prompted by some observations in sociophysics.

An extension of the Ising-Curie-Weiss model of self-organized criticality with a threshold on the interaction range

An extension of the Ising-Curie-Weiss model of self-organized criticality with a threshold on the interaction range

Self-Organized Criticality and Cognitive Control Reasoned by Effort Minimization

We put forward a novel model for self-organized criticality in the dynamics of systems controlled by human actions. The model is based on two premises. First, without human control, the system in issue undergoes supercritical instability. Second, the subject’s actions are aimed at preventing the occurrence of critical fluctuations when the risk of control failure becomes essential rather than keeping the system in the stability region. The latter premise is reasoned as follows: (i) keeping the system rather far from the instability boundary is not justified from the standpoint of effort minimization, and (ii) keeping it in the immediate proximity to the instability onset also requires considerable effort because of the bounded capacity of human cognition. The concept of dynamical traps is used in the mathematical description of this type of subject’s behavior. Numerical simulation demonstrates that the proposed model does predict the emergence of fluctuations with the power-law distribution. In conclusion, we discuss that the self-organized criticality of social systems is possible due to the basic features of the human mind.

On the Possibility of Reproducing Utsu's Law for Earthquakes with a Spring-Block SOC Model.

The Olami, Feder and Christensen (OFC) spring-block model has proven to be a powerful tool for analyzing and comparing synthetic and real earthquakes. This work proposes the possible reproduction of Utsu's law for earthquakes in the OFC model. Based on our previous works, several simulations characterizing real seismic regions were performed. We located the maximum earthquake in these regions and applied Utsu's formulae to identify a possible aftershock area and made comparisons between synthetic and real earthquakes. The research compares several equations to calculate the aftershock area and proposes a new one with the available data. Subsequently, the team performed new simulations and chose a mainshock to analyze the behavior of the surrounding events, so as to identify whether they could be catalogued as aftershocks and relate them to the aftershock area previously determined using the formula proposed. Additionally, the spatial location of those events was considered in order to classify them as aftershocks. Finally, we plot the epicenters of the mainshock, and the possible aftershocks comprised in the calculated area resembling the original work of Utsu. Having analyzed the results, it is likely to say that Utsu's law is reproducible using a spring-block model with a self-organized criticality (SOC) model.

Sandpile Universality in Social Inequality: Gini and Kolkata Measures.

Social inequalities are ubiquitous and evolve towards a universal limit. Herein, we extensively review the values of inequality measures, namely the Gini (g) index and the Kolkata (k) index, two standard measures of inequality used in the analysis of various social sectors through data analysis. The Kolkata index, denoted as k, indicates the proportion of the 'wealth' owned by (1-k) fraction of the 'people'. Our findings suggest that both the Gini index and the Kolkata index tend to converge to similar values (around g=k≈0.87, starting from the point of perfect equality, where g=0 and k=0.5) as competition increases in different social institutions, such as markets, movies, elections, universities, prize winning, battle fields, sports (Olympics), etc., under conditions of unrestricted competition (no social welfare or support mechanism). In this review, we present the concept of a generalized form of Pareto's 80/20 law (k=0.80), where the coincidence of inequality indices is observed. The observation of this coincidence is consistent with the precursor values of the g and k indices for the self-organized critical (SOC) state in self-tuned physical systems such as sand piles. These results provide quantitative support for the view that interacting socioeconomic systems can be understood within the framework of SOC, which has been hypothesized for many years. These findings suggest that the SOC model can be extended to capture the dynamics of complex socioeconomic systems and help us better understand their behavior.

Similar properties between gamma-ray emission of 3C 454.3 and solar GeV flares

ABSTRACTBased on the survey data observed by Fermi-Large Area Telescope (LAT), we elaborate the statistical characteristics of gamma-ray flares from 3C 454.3 and solar GeV flares among flare parameters, such as isotropic energy (Eγ), peak luminosity (LP), and duration time (TDuration). We find two significant correlations as: $T_{\rm Duration} \propto E_{\gamma }^{0.31\pm 0.03}$ and $L_{\rm P} \propto E_{\gamma }^{0.61\pm 0.03}$ for 3C 454.3. The exponents are in a better agreement those of the Sun from the first Fermi-LAT solar flare catalogue, namely $T_{\rm Duration,\odot } \propto E_{\gamma ,\odot }^{0.38\pm 0.08}$ and $L_{\rm P,\odot } \propto E_{\gamma ,\odot }^{0.81\pm 0.08}$. The relationship of TDuration–Eγ and LP–Eγ could be interpreted naturally as due to magnetic dissipation through reconnection. On top of that the frequency distributions of isotropic energy, peak luminosity, and duration time of gamma-ray emission of 3C 454.3 show power-law forms, and the waiting time distribution can be described by a non-stationary Poisson process. These distribution behaviours are comparable to those of the Sun, Sagittarius A*, and M87, and follow the prediction of a self-organized criticality model. All statistical properties suggest that similar to the physical process accounting for solar GeV flares and X-ray flares in supermassive black hole systems, magnetic reconnection cloud govern the energy-release, and particle acceleration process for gamma-ray flares of 3C 454.3.

Решение одномерной самоорганизованно-критической модели Осло

I study the Oslo model – a one-dimensional conservative strictly isotropic self-organized critical sandpile model. I obtain a complete analytical solution for it, and I also present confirming simulation results. The solution is based on a meso-level model. I consider the processes of this level as anomalous diffusion. To explain it, I consider various random walk mechanisms, one of which essentially takes into account the one-dimensional nature of the model.

Self-organized criticality in solar GeV flares

ABSTRACT The Sun emits significant flares in X-ray, ultraviolet, and radio wavelengths. It is thought to originate from the magnetic reconnection activity, which is capable of accelerating particles to high energies. The magnetic process can be described by the avalanche model of self-organized criticality (SOC), and it is evidenced by the observation. Here, we study the frequency distribution of fluence, peak flux, and duration time for solar GeV flares detected first by Fermi-Large Area Telescope. Their cumulative distributions show a power-law behaviour. The exponents are also consistent with those derived from the observations at low-energy bands, and follow the predictions of the fractal-diffuse SOC model. In the meantime, the waiting time shows power-law distribution, and agrees a non-stationary Poission process. We then explore the correlation between energy (fluence) and duration time using a two-variable regression analysis. The correlation is found to be $T_{\rm Duration} \propto F_{\rm GeV}^{0.38\pm 0.08}$ with the solar GeV flare sample, which is comparable to that of the solar X-ray flares and gamma-ray bursts (GRBs) and could be understood in an SOC model. These facts suggest that, similar to the physical process accounting for the X-ray emission of solar flares and prompt emission of GRBs, magnetic reconnection may still dominate the energy-release process and particle acceleration for solar flares at GeV energies.

Interface region imaging spectrograph (IRIS) observations of the fractal dimension in the solar atmosphere

We focus here on impulsive phenomena and Quiet-Sun features in the solar transition region, observed with the Interface Region Imaging Spectrograph (IRIS) at 1,400 Å (at formation temperatures of Te ≈ 104–106 K). Summarizing additional literature values we find the following fractal dimensions (in increasing order): DA = 1.23 ± 0.09 for photospheric granulation, DA =1.40 ± 0.09 for chromospheric (network) patterns, DA = 1.54 ± 0.04 for plages in the transition region, DA = 1.56 ± 0.08 for extreme ultra-violet (EUV) nanoflares, DA = 1.59 ± 0.20 for active regions in photospheric magnetograms, and DA = 1.76 ± 0.14 for large solar flares. We interpret low values of the fractal dimension (1.0 ≲ DA ≲ 1.5) in terms of sparse curvi-linear flow patterns, while high values of the fractal dimension (1.5 ≲ DA ≲ 2.0) indicate quasi-space-filling transport processes, such as chromospheric evaporation in flares. Phenomena in the solar transition region appear to be consistent with self-organized criticality (SOC) models, based on their fractality and their size distributions of fractal areas A and (radiative) energies E, which show power law slopes of αAobs=2.51±0.21 (with αAtheo=2.33 predicted), and αEobs=2.03±0.18 (with αEtheo=1.80 predicted). This agreement suggests that brightenings detected with IRIS at 1,400 Å reveal the same nonlinear SOC statistics as their coronal counterparts in EUV.

Success of social inequality measures in predicting critical or failure points in some models of physical systems

Statistical physicists and social scientists both extensively study some characteristic features of the unequal distributions of energy, cluster, or avalanche sizes and of income, wealth, etc., among the particles (or sites) and population, respectively. While physicists concentrate on the self-similar (fractal) structure (and the characteristic exponents) of the largest (percolating) cluster or avalanche, social scientists study the inequality indices such as Gini and Kolkata, given by the non-linearity of the Lorenz function representing the cumulative fraction of the wealth possessed by different fractions of the population. Here, using results from earlier publications and some new numerical and analytical results, we reviewed how the above-mentioned social inequality indices, when extracted from the unequal distributions of energy (in kinetic exchange models), cluster sizes (in percolation models), or avalanche sizes (in self-organized critical or fiber bundle models) can help in a major way in providing precursor signals for an approaching critical point or imminent failure point. Extensive numerical and some analytical results have been discussed.

Dimensional transmutation and nonconventional scaling behavior in a model of self-organized criticality

This paper addresses two unusual scaling regimes (types of critical behavior) predicted by the field-theoretic renormalization group analysis for a self-organized critical system with turbulent motion of the environment. The system is modeled by the anisotropic stochastic equation for a “running sandpile” introduced by Hwa and Kardar in [Phys. Rev. Lett. 62, 1813 (1989)]. The turbulent motion is described by the isotropic Kazantsev–Kraichnan “rapid-change” velocity ensemble for an incompressible fluid. The original Hwa–Kardar equation allows for independent scaling of the spatial coordinates [Formula: see text] (the coordinate along the preferred dimension) and [Formula: see text] (the coordinates in the orthogonal subspace to the preferred direction) that becomes impossible once the isotropic velocity ensemble is coupled to the equation. However, it is found that one of the regimes of the system’s critical behavior (the one where the isotropic turbulent motion is irrelevant) recovers the anisotropic scaling through “dimensional transmutation.” The latter manifests as a dimensionless ratio acquiring nontrivial canonical dimension. The critical regime where both the velocity ensemble and the nonlinearity of the Hwa–Kardar equation are relevant simultaneously is also characterized by “atypical” scaling. While the ordinary scaling with fixed IR irrelevant parameters is impossible in this regime, the “restricted” scaling where the times, the coordinates, and the dimensionless ratio are scaled becomes possible. This result brings to mind scaling hypotheses modifications (Stell’s weak scaling or Fisher’s generalized scaling) for systems with significantly different characteristic scales.

Universal predictability of large avalanches in the Manna sandpile model.

Substantiated explanations of the unpredictability regarding sandpile models of self-organized criticality (SOC) gave way to efficient forecasts of extremes in a few models. The appearance of extremes requires a preparation phase that ends with general overloading of the system and spatial clustering of the local stress. Here, we relate the predictability of large events to the system volume in the Manna and Bak-Tang-Wiesenfeld sandpiles, which are basic models of SOC. We establish that in the Manna model, the events located to the right of the power-law segment of the size-frequency relationship are predictable and the prediction efficiency is described by the universal linear dependence on the event size scaled by a power-law function of the lattice volume. Our scaling-based approach to predictability contributes to the theory of SOC and may elucidate the forecast of extremes in the dynamics of such systems with SOC as neuronal networks and earthquakes.

The Fractality and Size Distributions of Astrophysical Self-Organized Criticality Systems

The statistics of nonlinear processes in avalanching systems, based on the self-organized criticality (SOC) concept of Bak et al. (1988), predicts power-law-like size (or occurrence frequency) distribution functions. Following up on previous work, we define a standard SOC model in terms of six assumptions: (i) area fractality, (ii) volume fractality, (iii) the flux–volume proportionality, (iv) classical diffusion, (v) the Euclidean maximum at the event peak time, and (vi) the spatiotemporal fluence or energy of an avalanche event. We gather data of the fractal dimension and power-law slopes from 162 publications and assemble them in 28 groups (for instance, solar flare energies, or stellar flare energies), from which we find that 75% of the groups are consistent with the standard SOC model. Alternative SOC models (Lévy flight, flat-world, nonfractal) are slightly less correlated with the data. Outliers are attributed to small number statistics, background definition problems, inadequate fitting ranges, and deviations from ideal power laws.

Reconciling Power-law Slopes in Solar Flare and Nanoflare Size Distributions

We unify the power laws of size distributions of solar flare and nanoflare energies. We present three models that predict the power-law slopes α E of flare energies defined in terms of the 2D and 3D fractal dimensions (D A , D V ): (i) the spatiotemporal standard self-organized criticality model, defined by the power-law slope α E1 =1 + 2/(D V + 2) = (13/9) ≈ 1.44; (ii) the 2D thermal energy model, α E2 = 1 + 2/D A = (7/3) ≈ 2.33; and (iii) the 3D thermal energy model, α E3 = 1 + 2/D V = (9/5) ≈ 1.80. The theoretical predictions of energies are consistent with the observational values of these three groups, i.e., α E1 = 1.47 ± 0.07, α E2 = 2.38 ± 0.09, and α E3 = 1.80 ± 0.18. These results corroborate that the energy of nanoflares does not diverge at small energies, since (α E1 < 2) and (α E3 < 2), except for the 2D model (α E2 > 2). Thus, while this conclusion does not support nanoflare scenarios of coronal heating from a dimensionality point of view, magnetic reconnection processes with quasi-1D or quasi-2D current sheets cannot be ruled out.

A self-organized critical model and multifractal analysis for earthquakes in Central Alborz, Iran

This paper is devoted to a phenomenological study of the earthquakes in central Alborz, Iran. Using three observational quantities, namely the weight function, the quality factor, and the velocity model in this region, we develop a modified dissipative sandpile model which captures the main features of the system, especially the average activity field over the region of study. The model is based on external stimuli, the location of which is chosen (I) randomly, (II) on the faults, (III) on the low active points, (IV) on the moderately active points, and (V) on the highly active points in the region. We uncover some universal behaviors depending slightly on the method of external stimuli. A multi-fractal detrended fluctuation analysis is exploited to extract the spectrum of the Hurst exponent of the time series obtained by each of these schemes. Although the average Hurst exponent depends slightly on the method of stimuli, we numerically show that in all cases it is lower than 0.5, reflecting the anti-correlated nature of the system. The lowest average Hurst exponent is found to be associated with the case (V), in such a way that the more active the stimulated sites are, the lower the average Hurst exponent is obtained, i.e. the large earthquakes are more anticorrelated. Moreover, we find that the activity field achieved in this study provide information about the depth and topography of the basement, and also the area that can potentially be the location of the future large events. We successfully determine a high activity zone on the Mosha Fault, where the mainshock occurred on May 7th, 2020 (M_W 4.9).

Can the Auroral Kilometric Radiation be a Self‐Organized Criticality System?

AbstractWe present statistical results for the analysis of 240 Auroral Kilometric Radiation (AKR) dynamical spectra containing 3,750 short–time AKR bursts each, and recorded on‐board Interball–2 mission (POLRAD and MEMO experiments). Data were collected between December 1996 and February 1998. We found that frequency distribution of short AKR bursts as a function of their Power Spectral Density, shows for intense bursts a powerlaw fall. This is the hallmark of the Self–Organized Criticality (SOC) systems. In our case a simple analytical Logistic Growth SOC model can reproduce increments of Electron Cyclotron Maser instability expected for AKR.

Review and Update on Some Connections between a Spring-Block SOC Model and Actual Seismicity in the Case of Subduction Zones.

The self-organized critical (SOC) spring-block models are accessible and powerful computational tools for the study of seismic subduction. This work aims to highlight some important findings through an integrative approach of several actual seismic properties, reproduced by using the Olami, Feder, and Christensen (OFC) SOC model and some variations of it. A few interesting updates are also included. These results encompass some properties of the power laws present in the model, such as the Gutenberg-Richter (GR) law, the correlation between the parameters a and b of the linear frequency-magnitude relationship, the stepped plots for cumulative seismicity, and the distribution of the recurrence times of large earthquakes. The spring-block model has been related to other relevant properties of seismic phenomena, such as the fractal distribution of fault sizes, and can be combined with the work of Aki, who established an interesting relationship between the fractal dimension and the b-value of the Gutenberg-Richter relationship. Also included is the work incorporating the idea of asperities, which allowed us to incorporate several inhomogeneous models in the spring-block automaton. Finally, the incorporation of a Ruff-Kanamori-type diagram for synthetic seismicity, which is in reasonable accordance with the original Ruff and Kanamori diagram for real seismicity, is discussed.

Near universal values of social inequality indices in self-organized critical models

We have studied few social inequality measures associated with the sub-critical dynamical features (measured in terms of the avalanche size distributions) of four self-organized critical models while the corresponding systems approach their respective stationary critical states. It has been observed that these inequality measures (specifically the Gini and Kolkata indices) exhibit nearly universal values though the models studied here are widely different, namely the Bak–Tang–Wiesenfeld sandpile, the Manna sandpile and the quenched Edwards–Wilkinson interface, and the fiber bundle interface. These observations suggest that the self-organized critical systems have broad similarity in terms of these inequality measures. A comparison with similar earlier observations in the data of socio-economic systems with unrestricted competitions suggest the emergent inequality as a result of the possible proximity to the self-organized critical states.